Problem 46 - Algebra for College Students

Question

Simplify each expression. Express final results without using zero or negative integers as exponents. \(b^{-2} \cdot b^{3} \cdot b^{-6}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{b^{5}} \).

Step 1: Apply the Product of Powers Rule

The Product of Powers Rule states that when multiplying like bases, we can add the exponents together. Here, we have the expression: \[ b^{-2} imes b^{3} imes b^{-6} \]We add the exponents: \[ -2 + 3 - 6 \]

Step 2: Simplify the Exponents

Now, perform the addition of the exponents: \[ -2 + 3 - 6 = -5 \] Thus, the expression becomes:\[ b^{-5} \]

Step 3: Convert Negative Exponent to Positive

To express the expression with a positive exponent, recall that a negative exponent indicates a reciprocal. Hence, \[ b^{-5} = \frac{1}{b^{5}} \] This conversion removes the negative exponent by moving the base to the denominator.

Key Concepts

Product of Powers Rule

Whenever you encounter an expression involving powers of the same base being multiplied, the Product of Powers Rule is your go-to tool. This rule is simple and yet very powerful: when you multiply expressions with the same base, you add their exponents. In this example, the expression given is \[ b^{-2} \cdot b^{3} \cdot b^{-6} \]So, you would add the exponents together like this:\[ -2 + 3 - 6 \]By adding these numbers, you're really just simplifying the expression down to one with a single power of the base, making it a lot easier to manage.

Negative Exponents

Negative exponents might seem tricky, but they're quite easy to understand with a bit of practice. A negative exponent indicates a reciprocal. For example, if you have a negative exponent like in the expression \( b^{-5} \), you need to transform it into a positive exponent by using the reciprocal. So, this becomes:\[ b^{-5} = \frac{1}{b^5} \] This conversion helps us deal with expressions using positive exponents, which are typically easier to interpret and understand. Remember, moving the base with its exponent to the denominator is how we change a negative exponent into a positive one, and vice versa if needed.

Simplifying Expressions

Simplifying expressions is all about making mathematical expressions as compact and clear as possible. In the given example, the original complex expression with multiple powers of \( b \) is transformed through the use of exponent rules.

• First, use the Product of Powers Rule to combine the powers.
• Then, convert any negative exponents by understanding them as reciprocals.

These steps simplify the expression from \( b^{-2} \cdot b^{3} \cdot b^{-6} \) to the much simpler \( \frac{1}{b^5} \). Using the exponent rules not only makes your work cleaner, but it also reduces the chances of making errors in complex calculations.